I think perhaps you are worrying too much and she is playing on your own fears, especially about your own childhood.
There's nothing wrong with playing in the bath and enjoying the story when you read it to her. I don't know any child that falls asleep midway through a story (other than in films). My son plays in the bath and is interested and excited by the stories I read. He still goes to sleep (getting later now but he still needs 10 hours).
I found the best way when he was refusing to sleep and having a tantrum was to put a nursery rhyme CD on and leave him too it. Yes he did get up and switch it on again sometimes, but he eventually went off to sleep, and with less tantrums than before. I also leave a nightlight on so he can see around his room, but not enough to read by.
Now he's 9 and his CD at bedtime is a story such as horrid henry or charlie and the cholcoate factory. I also let him read his books in bed for half an hour before I come up and switch the main light off and tuck him in. He then listens to his CD and falls asleep.
If your daughter doesn't want to go straight to sleep leave her to 'read' her books.
My son is meant to go to bed at 7.30pm and lights out at 8pm. Then he's up around 7am. However I do let him play out in the garden a bit later in the summer. I would rather he was doing that than watching tv. But he needs the same amount of sleep so the later he goes to bed the later he gets up.
We've had our fair share of tantrums and getting up all the time for any excuse; still hungry, glass of water, can i sleep in your bed, not tired, who's on the phone... You just have to be firm and say it's bed time now.
Yes your daughter will be very tired when she starts school. Some reception year children fall asleep at their desk even! But you can do things now to wear her out physically as well as mentally.
Do you have an indoor play center near you? She's old enough now that you can let her run round on her own and chill out with a cup of coffee. Do you take her out to the park or for a long walk? Even a 20 minute walk when you get home can be enough to tire her out and will hopefully reap rewards by cutting down the tantrums at bed time.
Don't carry her when she says she is tired. Don't leave her to sit watching tv while you cook dinner. Do you have an outdoor area with some play equipment? Send her out to play on that if you can see her while you cook dinner.
And ask the staff at nursery to let you know if and when she has a nap. And if she falls asleep after 2pm ask that they wake her after 15 minutes as you want her to sleep in the evening, but also because she won't be able to sleep at school and needs to get out of an afternoon nap.
My last boyfriend's daughter wouldn't settle at night when she stayed with him. However one day at mine when we took her to the beach and she walked a long way and run around and climbed the sand dunes and built castles and got lots of fresh air she went to bed and fell almost right to sleep. She was in bed by 7.30, has 3 stories and was fast asleep 10 minutes after we left her! He was amazed. But that was what fresh air, stimulation and exercise did. At his place she was pushed everywhere in the buggy because he was in a rush.
I do know how exhausting lack of sleep and lack of you time can be. Take time to chill and relax and try not to get too stressed about it. Easier said than done I know.
Best wishes
Gilly